Initial-boundary value and inverse problems for subdiffusion equations in $\mathbb{R}^N$

Abstract

An initial-boundary value problem for a subdiffusion equation with an elliptic operator $A(D)$ in $\mathbb{R}^N$ is considered. The existence and uniqueness theorems for a solution of this problem are proved by the Fourier method. Considering the order of the Caputo time-fractional derivative as an unknown parameter, the corresponding inverse problem of determining this order is studied. It is proved, that the Fourier transform of the solution $\hat{u}(\xi, t)$ at a fixed time instance recovers uniquely the unknown parameter. Further, a similar initial-boundary value problem is investigated in the case when operator $A(D)$ is replaced by its power $A^\sigma$. Finally, the existence and uniqueness theorems for a solution of the inverse problem of determining both the orders of fractional derivatives with respect to time and the degree $ \sigma $ are proved. We also note that when solving the inverse problems, a decrease in the parameter $\rho$ of the Mettag-Leffler functions $E_\rho$ has been proved.